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3 Pole and shape determination


In order to determine the orientation of the rotational axis and the shape of the asteroid, we used the amplitude-magnitude (AM) method suggested by Zappalà (1981) and refined by Zappalà et al. (1983a), which is based on the assumed ellipsoidal shape of the asteroid (with semi-axes a > b > c) and on the relationships between the aspect angle, the lightcurve amplitude and the asteroid magnitude at the lightcurve maximum, all obtained in several oppositions (at least three). It is important to note that usually the real shape of asteroids is different from the ellipsoidal one and moreover the albedo is often not homogeneous over the entire surface. These discrepancies often lead to conflicting results especially when the data are few.
From the lightcurves, we obtain the magnitude V at the maximum of lightcurve and the amplitude A, depending on the rotation axis orientation and on the ratio of the maximum to minimum cross-sections of the asteroid, respectively.
If we assume the smaller axis c to be the asteroid rotation axis, the ratio between the two other axes, and subsequently their single values, can be obtained from the $(A-\lambda)$ plot, if we have a continuous and good distribution in longitude of the observed amplitudes. The modelling curves were obtained using the least square method.
In some cases the extrema of the theoretical curves seem to be overestimated with respect to the observed values. This fact depends on the computing program that, in the absence of observed values at the longitudes of the maximum or the minimum, takes into account the slope of the ascending or descending branches.
From the axes ratios it is possible to obtain the value of the aspect angle (with an uncertain definition of the north or south pole) and hence the pole longitude.

  
Table 1: References of the lightcurves used for the estimation of the V magnitudes and for the construction of the $(A-\lambda)$ plots. The symbol before the author's name is the same used in the corresponding $(A-\lambda)$ plots
\begin{table*}
\begin{center}
{
\psfig {file=ds4996f7.eps,width=14cm,height=18cm}
}\end{center}\vspace*{-10mm}\end{table*}


 
Table 1: continued
\begin{table*}
\begin{center}
{
\psfig {file=ds4996f6.eps,width=12cm,height=18cm}
}\end{center}\vspace*{-10mm}\end{table*}

Following Zappalà et al. (1990) suggestions, we have corrected the lightcurve amplitude for its dependence on the phase angle, by means of the relationship $A(0^{\circ})=\frac{A(\alpha)}{(1+m\alpha)}$where $A(\alpha)$ is the observed lightcurve amplitude, $\alpha$ is the solar phase angle and m is a coefficient depending on the asteroid taxonomic class. The $V_{0}(1,\alpha)$ of each asteroid was computed adopting the value $\alpha_m$, the arithmetic average of all phase angles.

In Table 1, for each asteroid, the references of the lightcurves used for the estimation of the V magnitude and for the construction of the $(A-\lambda)$ plots are reported. The symbol before the author's name is that used in the corresponding $(A-\lambda)$ plots. Only lightcurves at least 90% covered were utilized. Due to the available lightcurves, their minimum number (at least three) necessary for applying the (AM) method and to their distribution in longitude, it was possible to compute the pole coordinates and the axes ratios only for 30 asteroids. In Fig. 1, using different symbols for different authors as indicated in Table 1, the $(A-\lambda)$ plots of these asteroids are reported. The $\lambda$ adopted values are the mean values computed over the duration of each lightcurve. The filled symbols indicate the observed values of the amplitude A, the empty ones the corresponding values at longitudes $\lambda + 180^{\circ}$, the continuous and dashed (in the case of two solutions) lines the theoretical curves.

  
\begin{figure}
\begin{center}
\includegraphics[width=16cm,height=12cm,clip=]{ds4...
 ...e=ds4996f2.ps,width=16cm,height=12.cm}
 \end{center}\vspace*{-2.5cm}\end{figure} Figure 1: Amplitude-longitude plots of the asteroid to which it was possible to apply the (AM) method. The filled symbols, as reported in Table 1, indicate the observed values of the amplitude A, the empty ones the corresponding ones at longitudes $\lambda + 180^{\circ}$, the continuous and (in the case of two solutions) dashed lines the theoretical curves

 
\begin{figure}
\centering{
 
\psfig {figure=ds4996f3.ps,width=16cm,height=12.5cm...
 ...sfig {figure=ds4996f4.ps,width=16cm,height=12cm}
}
\vspace*{-2.5cm} \end{figure} Figure 1: continued

 
\begin{figure}
\centering{
 
\psfig {figure=ds4996f5.ps,width=16cm,height=12.5cm}
}
\vspace*{-2.5cm} \end{figure} Figure 1: continued

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